= Solution
The change to <ingoing Kerr coordinates> adds functions of $r$ to $t$ and $\phi$, and leaves $r,\theta$ unchanged. Differentiating at fixed $r,\theta,\phi$ therefore gives $\partial_t v=1$ and $\partial_t\chi=0$. Differentiating at fixed $r,\theta,t$ gives $\partial_\phi\chi=1$ and $\partial_\phi v=0$. Hence the two <Killing vector fields> are
$$
\boxed{k=\partial_v,\qquad m=\partial_\chi.}
$$
They remain the stationary and axial <Killing vector fields>; the coordinate change does not mix their generators with $\partial_r$.
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